Numerical analysis homework help

MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 1 of 3
(Show all work.) [1] Find the small positive integer solution of each problem: (38 pts)
(Hint: you may find any integer solution first, then add/subtract a multiple of n)
[1a] 7x ≡ 6 (mod 41), 0 ≤ ?? ≤ 40
 
[1b] 35x ≡ 15 (mod 64), 0 ≤ ?? ≤ 64
[1c] 0 ≤ ?? ≤ 17 • 19 = 323
� x ≡ 5 (mod 17) x ≡ 11 (mod 19)
 
 
 
 
 
 
 
 
 
 
 

 
MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 2 of 3
[2] Given ?? = 41 ?????? ?? = 9. Solve each equation: (20 pts) [2a] Find ?? satisfying 1 ≤ ?? < 41 and ???? ≡ 1 (?????? 41)
 
[2b] Find ?? satisfying 1 ≤ ?? < 41 and ???? ≡ 28 (?????? 41) (Use [2a] is easier.)
[3] Find the set of quadratic residues and the set of nonresidues of p = 17. 12 pts
 

 
MATH 312 Final Exam Past Due Date Final Exam Will Not Be accepted
Due on Sunday, December 06, 2020 Print your name: _________________________________.
Page 3 of 3
[4] Use [4*] to find the smallest positive integer solutions (if any) of each quadratic congruence equation: (30 pts)
[4*] 12 ≡ 1 (?????? 11), 22 ≡ 4 (?????? 11), 32 ≡ 9 (?????? 11), 42 ≡ 5 (?????? 11),
52 ≡ 3 (?????? 11), 62 ≡ 3 (?????? 11), 72 ≡ 5 (?????? 11), 82 ≡ 9 (?????? 11),
92 ≡ 4 (?????? 11), 102 ≡ 1 (?????? 11)
[4a] Solve the congruence equation: 9??2 + 6?? + 1 ≡ 0(?????? 11), 0 ≤ ?? ≤ 10
 
 
 
 
 
 
[4b] Solve the congruence equation: 5??2 + 3?? + 2 ≡ 0 (?????? 11), 0 ≤ ?? ≤ 10
 
 
 
 
 
[4c] Solve the congruence equation: 3??2 − 4?? + 1 ≡ 0 (?????? 11), 0 ≤ ?? ≤ 10